Gauss-Prym maps on Enriques surfaces

Abstract

We prove that the $k$-th Gaussian map ${\gamma }_H^k$ is surjective on a polarized unnodal Enriques surface $(S,H)$ with $\varphi (H)>2k+4$. In particular, as a consequence, when $\varphi (H)>4(k+2)$, we obtain the surjectivity of the $k$-th Gauss-Prym map ${\gamma }_{{\omega }_C\otimes \alpha }^k$ on smooth hyperplane sections $C\in |H|.$ In case $k=1$ it is sufficient to ask $\varphi (H)>6$.

Fixed some typos and improved the exposition of some parts from the first version